Partial shading conditions (PSCs) caused by uneven illumination become one of the most common problems in photovoltaic (PV) systems, which can make the PV power-voltage (P-V) characteristics curve show multi-peaks. Traditional maximum power point tracking (MPPT) methods have shortcomings in tracking to the global maximum power point (GMPP), resulting in a dramatic decrease in output power. In order to solve the above problems, intelligent algorithms are used in MPPT. However, the existing intelligent algorithms have some disadvantages, such as slow convergence speed and large search oscillation. Therefore, an improved whale algorithm (IWOA) combined with the P&O (IWOA-P&O) is proposed for the MPPT of PV power generation in this paper. Firstly, IWOA is used to track the range interval of the GMPP, and then P&O is used to accurately find the MPP in that interval. Compared with other algorithms, simulation results show that this method has an average tracking efficiency of 99.79% and an average tracking time of 0.16 s when tracking GMPP. Finally, experimental verification is conducted, and the results show that the proposed algorithm has better MPPT performance compared to popular particle swarm optimization (PSO) and PSO-P&O algorithms.

With the rapid consumption of traditional energy sources such as fossil fuels and the worsening of environmental pollution, the human living environment has been severely damaged. Governments around the world have called for the development and utilization of clean energy, so solar energy, as a clean and renewable energy source, has received widespread attention from people [

In order to track GMPP under PSCs, many researchers have proposed to apply intelligent algorithms to MPPT, the most common of which is particle swarm optimization (PSO) [

Although the above algorithms can realize MPPT, the principle is complex. Whale algorithm (WOA) is a new optimization algorithm proposed by Mirjalili et al. based on the behavior of whale prey hunting [

The structure of the paper is as follows.

The PV array is composed of multiple PV cells in series and parallel. The equivalent model of the PV cell is shown in _{ph} is photogenic current; _{D} is diode current; _{p} is the bypass resistance; _{S} is the internal series resistance; _{pv} is the PV output voltage; _{pv} is the output current.

The output voltage-current characteristics of the PV cell can be expressed as:
_{o} is the reverse saturation current;

In this paper, the PV cell (A10J-M60-240) is used as the research object. The parameters of this PV cell under a standard environment (temperature is 25°C and irradiance is 1000 W/m^{2}) are shown in _{oc} is the open circuit voltage; _{sc} is the short circuit current; _{m} is the voltage of MPP; _{m} is the current of MPP; _{max} is the maximum power.

Parameter | Value | Parameter | Value |
---|---|---|---|

_{oc} |
36.84 V | Temperature coefficient of _{oc}(%/deg.C) |
−0.359 |

_{sc} |
8.32 A | Temperature coefficient of _{sc}(%/deg.C) |
0.097 |

_{m} |
30.72 V | _{max} |
240.5 W |

_{m} |
7.83 A |

Due to the wide distribution of PV arrays, in actual working conditions, haze, dust, trees, or buildings will lead to uneven illumination, resulting in PSC. The above three PV cells are connected in series into a PV array as the simulation model. PV array is shown in

In order to fully demonstrate the influence of illumination on PV output, different irradiances are set. Detailed data on irradiances are shown in _{1}, S_{2}, and S_{3} respectively represent the solar irradiance on PV1, PV2, and PV3. The power-voltage (P-V) and current-voltage (I-V) curves under different PSCs are illustrated in

Condition | S_{1} (W/m^{2}) |
S_{2} (W/m^{2}) |
S_{3} (W/m^{2}) |
---|---|---|---|

PSC1 | 1000 | 1000 | 1000 |

PSC2 | 1000 | 1000 | 800 |

PSC3 | 900 | 700 | 400 |

As is shown in ^{2} (PSC1), the P-V curve of the PV array has only one peak, and the I-V curve is a single-knee shape. In the other two cases, the irradiance of the three PV cells in the array is uneven. In PSC2, three PV cells are exposed to two irradiances (1000 W/m^{2} for S_{1} and S_{2}, 800 W/m^{2} for S_{3}), and the P-V curve has two peaks, and the I-V curve turns into double-knee shape. In PSC3 (900 W/m^{2} for S_{1}, 700 W/m^{2} for S_{2}, 400 W/m^{2} for S_{3}), the P-V curve has three peaks, and the I-V curve turns into three-knee shape.

When WOA is applied to the MPPT, the position of each whale corresponds to the output voltage. The minimum voltage of the PV array corresponds to the lower bound of the whale’s hunting range, and the maximum voltage corresponds to the upper bound of the whale’s hunting range. The whales in the search range will change their positions according to equations until the position of MPP is found, and then the search will stop after the voltage is output. It consists of three modes: encircling prey, bubble-net hunting technique, and random searching for prey [

During this phase, the position of the whale corresponding to the voltage will locally move according to the following equation:

where,

where, _{1} and _{2} are random parameters in [0,1]; _{max} is the maximum number of iterations.

At this stage, the position of the whale corresponding to the voltage moves in a spiral toward the optimal value. The equation for updating the position is as follows:

The mathematical model of this stage is as follows:

where, _{rand}(

The P&O algorithm is a traditional MPPT algorithm. The principle of the P&O is to compare

In the actual function optimization, the change rate of _{1} and the weight factor

By using _{1}, and

The variable _{1} with nonlinear change is adopted. According to _{1} in IWOA has a significant rate of change in the early stages of iteration, resulting in the algorithm having the largest search range and changing the search step size within a larger range. The IWOA ensures global search ability while also considering search speed. The value of the parameter _{1} decreases rapidly with the number of iterations in the later stage, and the search range of IWOA becomes smaller. Whale individuals converge around the optimal value. Finally, the IWOA obtains the optimal value with a small range of fluctuations.

At the end of the algorithm, the variance of the position (corresponding to voltage) of the whale population and setting the variance threshold are used to determine whether the whale is close to the maximum power point (MPP). When the variance of the location of the the whale population is less than variance threshold, the whale is located near the MPP. The threshold affects the tracking performance of the algorithm. If the threshold is too large, the algorithm will prematurely converge to other local maximum power point (LMPP), resulting in the algorithm tracking the wrong value. Otherwise, if the threshold is too small, the number of iterations will increase and the rate of convergence will decrease. Finally, the voltage corresponding to the position of the optimal whale is taken as the initial point of P&O to finely track the MPP of the PV array. The discriminant near the MPP is as follows:

^{2} is the variance of the position of whale population. _{i} is the voltage corresponding to the position of the

If the external illumination changes, the MPP of the PV array will change. At this point, the algorithm needs to be restarted to track the MPP of the PV array in real time. The restart condition is as follows:

The specific MPPT process of IWOA-P&O is as follows:

(1) Firstly, the maximum iteration number _{max}, the population number

(2) Secondly, the voltage and current of PV array are collected, and the power calculation formula is set as the objective function. The MPP is the target that the whale finally seeks.

(3) Then the convergence factor _{1} and the control parameter _{1} are updated according to

(4) At the same time, the output power corresponding to each whale position is calculated, and the optimal individual power is taken as the historical global optimal value. If the current population optimum value is greater than the historical global optimal value, the current population optimum is regarded as the new optimum, and the historical global optimum is updated. Otherwise, the historical optimal value is retained, and then the voltage corresponding to the position of the whale is updated continuously.

(5) On this basis, when

(6) Then the voltage corresponding to the position of the optimal individual in the whale population is taken as the initial point of P&O, and the P&O is carried out in a small step size until the steady state is reached.

(7) Finally, if

In order to verify the superiority of the IWOA-P&O proposed in this paper, three PV cells (A10J-M60-240) combined with Boost circuits are used for simulation experiments. The photovoltaic (PV) power generation system is set up as shown in _{1} is 200 uF; The busbar capacitance _{2} is 180 uF; The inductance ^{2}, respectively. PSC2: The illuminations received by the three PV cells are 1000, 1000, and 800 W/m^{2}, respectively. PSC3: The illuminations received by the three PV cells are 900, 700, and 400 W/m^{2}, respectively. PSC4: The illuminations received by the three PV cells are 500, 300, and 100 W/m^{2}, respectively. The power-voltage (P-V) curves of four cases are shown in

Simulations are performed under static PSCs. The MPPT results of different algorithms under PSC1 are shown in

The MPPT results of different algorithms under PSC2 are shown in

The MPPT results of different algorithms under PSC3 are shown in

The MPPT results of different algorithms under PSC4 are shown in

Condition | Algorithm | Tracking time (s) | Output power (W) | Efficiency (%) | GMPP tracked |
---|---|---|---|---|---|

WOA | 0.23 | 720 | 99.82 | Yes | |

PSO | 0.3 | 720 | 99.82 | Yes | |

PSC1 | P&O | 0.2 | 721 | 99.96 | Yes |

PSO-P&O | 0.11 | 721 | 99.96 | Yes | |

IWOA-P&O | 0.1 | 721 | 99.96 | Yes | |

WOA | 0.29 | 563.4 | 91.12 | No | |

PSO | 0.4 | 617 | 99.79 | Yes | |

PSC2 | P&O | 0.19 | 474.5 | 76.74 | No |

PSO-P&O | 0.21 | 618 | 99.95 | Yes | |

IWOA-P&O | 0.18 | 618 | 99.95 | Yes | |

WOA | 0.28 | 348 | 99.54 | Yes | |

PSO | 0.38 | 345 | 98.68 | Yes | |

PSC3 | P&O | 0.19 | 318 | 90.96 | No |

PSO-P&O | 0.22 | 348 | 99.54 | Yes | |

IWOA-P&O | 0.14 | 349.3 | 99.91 | Yes | |

WOA | 0.29 | 114 | 76 | No | |

PSO | 0.4 | 141 | 94 | Yes | |

PSC4 | P&O | 0.17 | 78 | 52 | No |

PSO-P&O | 0.26 | 149 | 99.33 | Yes | |

IWOA-P&O | 0.21 | 149 | 99.33 | Yes |

The superiority of IWOA-P&O algorithm has been verified under static PSCs. However, in the real world, illumination is not invariable and will change with time. For this reason, a variation of PSC is performed which is changed at 0.35 and 0.7 s in the order of PSC1-PSC3-PSC2. The P-V curve changes from a single peak to a triple peak and then to a double peak.

The MPPT results of WOA, PSO, P&O, and IWOA-P&O algorithms are shown in

Stage | Algorithm | Tracking time (s) | Output power (W) | Efficiency (%) | GMPP tracked |
---|---|---|---|---|---|

WOA | 0.23 | 721 | 99.96 | Yes | |

PSO | 0.32 | 720 | 99.82 | Yes | |

[0, 0.35 s] | P&O | 0.2 | 721 | 99.96 | Yes |

PSO-P&O | 0.12 | 721 | 99.96 | Yes | |

IWOA-P&O | 0.1 | 721 | 99.96 | Yes | |

WOA | 0.05 | 304.3 | 87.04 | No | |

PSO | 0.05 | 304 | 86.96 | No | |

[0.35 s, 0.7 s] | P&O | 0.25 | 318.2 | 91.02 | No |

PSO-P&O | 0.04 | 307 | 87.8 | No | |

IWOA-P&O | 0.12 | 349.3 | 99.91 | Yes | |

WOA | 0.05 | 606.5 | 98.09 | No | |

PSO | 0.06 | 604 | 97.69 | No | |

[0.7 s, 1 s] | P&O | 0.21 | 618 | 99.95 | Yes |

PSO-P&O | 0.01 | 611 | 98.82 | Yes | |

IWOA-P&O | 0.17 | 618 | 99.95 | Yes |

To verify the MPPT performance of the proposed method in series and parallel connection of multiple photoelectric cells, nine PV cells are connected, as shown in ^{2}. The illumination received by PV4, PV5 and PV6 is 800 W/m^{2}. The illumination received by PV7, PV8 and PV9 is 600 W/m^{2}. The P-V curve of the PV array is shown in

The continuously varying irradiance and temperature are simulated to verify the MPPT performance of the proposed method. As shown in ^{2} at 0.2 s and reaches the lowest point of 600 W/m^{2} at 0.4 s. Then irradiance rises and returns to 1000 W/m^{2} at 0.9 s. The theoretical MPP of the PV array and results of IWOA-P&O MPPT are shown in

In addition, in order to verify the MPPT performance of the proposed method, the simulation results of the dynamic process under continuously changing PSCs are shown in

To validate the proposed MPPT method, a boost circuit experimental platform is built in the lab. The photo of the platform is shown in

Three different P-V curves are set up in the experiment to validate different algorithms, which are shown in

Aiming at the multi-peak characteristics of PV systems under PSCs, IWOA-P&O is proposed in this paper to realize MPPT. The convergence factor and weight factor are introduced in IWOA-P&O, and combined with P&O of strong local search ability, so the convergence accuracy and speed of MPPT can be effectively improved. The simulations of the PV power generation system are built in MATLAB/Simulink. Simulation results show that Compared with WOA, PSO, P&O, and PSO-P&O, the convergence time of IWOA-P&O is all within 0.21 s, the average convergence time is reduced by more than 50%, and the average efficiency of MPPT reaches 99.79%. In addition, in the case of illumination variation, IWOA-P&O can restart, the convergence speed is faster, and the tracking efficiency is higher. Finally, experimental verification is conducted. Experimental results show that the proposed method is feasible, avoids the disadvantage of falling into the LMPP, reduces power loss, and has good stability.

The authors would like to express their gratitude to the reviewers and the editor.

This work was supported in part by the Natural Science Foundation of Jiangsu Province under Grant BK20200969 (L. Z., URL:

The authors confirm contribution to the paper as follows: study conception and design: Jian Zhong, Lei Zhang, Ling Qin; data collection: Ling Qin; analysis and interpretation of results: Jian Zhong, Lei Zhang; draft manuscript preparation: Jian Zhong. All authors reviewed the results and approved the final version of the manuscript.

Data supporting this study are included within the article.

The authors declare that they have no conflicts of interest to report regarding the present study.